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mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
RAND Journal of Economics
Vol. 36, No. 3, Autumn 2005
pp. 520–543
Storability, market structure, and
demand-shift incentives
James J. Anton∗
and
Gopal Das Varma∗∗
We consider a two-period model in which buyers can store a good by purchasing in advance of
consumption so as to realize potential gains from intertemporal arbitrage. We find that storability
introduces a kink in the aggregate period-1 demand. When supply is oligopolistic (quantity setting)
and consumers are sufficiently patient (storage cost is relatively low), each firm has a strong current
incentive to capture future market share from a rival. As a result, in equilibrium, the price path
is increasing and there is rational in-advance purchase by buyers. In contrast, the monopoly and
perfectly competitive markets exhibit no such price dynamics. Intermediate storage costs result
in multiple equilibria, with at least one that involves advance purchase and one that does not.
1. Introduction
We frequently observe sellers offering a storable good at reduced prices relative to those in
past or future market periods. Moreover, these price variations are often anticipated by buyers of
the good. Familiar examples include weekly supermarket advertisements that encourage buyers to
“stock up” on items and the common practice of having a “sale” at the same points in time across
the year. In addition to consumers and retail markets, we also observe manufacturers in wholesale
markets who offer incentives to purchase in advance. Retailers often take advantage of the low
prices offered by a manufacturer during a trade-deal promotion to engage in forward buying and
build inventory (see Blattberg and Neslin, 1990). This raises the question of why sellers would
want to shift the sales of goods consumed repeatedly by buyers to periods in which prices are
relatively low.
In this article we argue for the necessity of price dynamics in the market for a storable
good. When supply is imperfectly competitive, we find that market outcomes are characterized
by endogenous dynamics even when the underlying cost and demand structure is unchanging
∗
Duke University; [email protected].
CRA International; [email protected].
For helpful comments, we are grateful to the Editor, two anonymous referees, Gary Biglaiser, Judy Chevalier,
Tracy Lewis, Alessandro Lizzeri, David McAdams, Martin Pesendorfer, Dennis Yao, and participants at the Northwestern
2003 CSIO Conference and seminars at Arizona, Duke, Indiana, Penn, UBC, Vanderbilt, Virginia Tech, and Wisconsin.
Financial support from the Business Associates Fund at the Fuqua School of Business is gratefully acknowledged. The
ideas and opinions expressed in this article are solely those of the authors and do not necessarily represent the views of
CRA International and its staff.
∗∗
520
Copyright © 2005, RAND.
mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
ANTON AND DAS VARMA / 521
over time. The economic force behind this result is that an oligopolistic (quantity-setting) firm
has a strong incentive to shift demand for a storable good from the future, where it is shared with
rivals, into the present by inducing buyers to store the good.1 In contrast, demand shifting does not
arise under perfect competition or pure monopoly, where a repetition of the static single-period
equilibrium is stable. Because oligopolistic competitors find it profitable to capture future market
share from rivals, the single-period outcome is unstable. Instead, the market has an initial highly
competitive or low-price phase in which buyers store the good and then a high-price phase in
which buyers draw down inventories for consumption.2
Recent empirical work by Erdem, Imai, and Keane (2003) and Hendel and Nevo (2005)
focuses on identifying the extent of buyer storage by examining dynamic structural models of
consumer demand. Both find that buyers are forward-looking and respond when the current price
is low relative to expected future prices. Furthermore, they report that static estimates, which
ignore dynamic components, overstate the price elasticity of consumption demand (by at least a
factor of 2).
The empirical distinction between static and dynamic price elasticities is helpful for understanding the equilibrium structure and predictions of our model. In Figure 1 we exhibit the static
consumption demands, which we take for now to be the same in each period. Incorporating storage
demand, which is derived as part of the equilibrium, we have the solid lines in Figure 1 (linearity is
convenient for the graph; the model requires only downward-sloping marginal revenue). Demand
in period 1 coincides with consumption demand until the price is low enough to trigger storage
by buyers (the kink at K). Demand in period 2 is then shifted in from consumption demand, much
like a residual-demand curve.
If the good were not storable, the equilibrium would be at point C (static Cournot outcome)
in each period and no dynamics arise. When the good is storable, however, the equilibrium is
at A in period 1 and then B in period 2, with prices rising and quantity purchased falling. In
period 1, we have a low price and buyers are induced to store the good in the amount x as firms
compete aggressively, reflecting the demand-shift incentive. Because of storage, residual demand
in period 2 is shifted inward from the static consumption demand. Calculating a simple price
elasticity, (q A − q B )/q B divided by ( p A − p B )/ p B , we see that a static interpretation overstates
the elasticity for two reasons. First, demand at A includes a storage component, which means q A is
above the static level for period 1. Second, the demand q B at B includes a residual-demand effect
and is thus below the static level for period 2. Both effects work toward providing an equilibrium
account that is consistent with the dynamic empirical demand elasticity findings noted above.
Our basic model is a standard two-period quantity-setting duopoly market in which we
introduce only one structural economic change: the good is storable by the buyers between
periods. The firms sell a homogeneous output, which avoids the complication of differentiation
issues, and produce at a constant marginal cost. Each buyer seeks to consume a single unit of the
good in each period. Buyers are heterogeneous in their valuation of the good, and valuations satisfy
the decreasing inverse hazard rate property (a similar demand structure arises for homogeneous
consumers with concave utility over continuous quantities). This implies decreasing marginal
revenue and, consequently, the static (one-period) version of the game has a unique equilibrium
at the standard Cournot outcome.
Buyers are allowed to store the good (at no direct cost) by purchasing today for future
consumption. All agents are risk neutral and discount future payoffs at the same rate. While only
the firms have market power and buyers act as price takers, we want to capture the notion that
each buyer makes an optimal storage decision relative to an accurate prediction of how prices will
1 A similar incentive arises for price setting with differentiated products; quantity setting allows us to study the
simpler case of a homogeneous good. With differentiated products, we must keep track of storage for each good, and
storage behavior by the buyers becomes more intricate as it involves comparisons of surplus for each good across different
prices (leading to regions rather than simple inequalities to determine when storage occurs for one or more of the goods).
2 Demand shifting also arises with durable goods and with forward contracting, as discussed later in the Introduction.
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FIGURE 1
ENDOGENOUS DEMANDS
change over time. Thus, buyers have rational expectations (perfect foresight) regarding equilibrium
prices and storage demand is positive only when the current price is below the discounted expected
price of the good next period. We also examine comparative statics with respect to demand growth
or contraction.
The equilibrium exhibits three important properties: the demand-shift incentive, price arbitrage restrictions generated by storage demand, and the resulting intertemporal price and quantity
variation. To gain intuition, consider a repetition of the static Cournot outcome. This is, in fact, a
local profit maximum for each firm, and buyers have a strict incentive not to store the good, since
the price is expected to remain constant. In Figure 1, point C is a discrete distance from the kink
where storage demand becomes positive. This outcome, however, is not a global best response
for either firm.
A sufficiently large deviation from the Cournot output will drive the current price down and
trigger storage by buyers. Although this depresses prices, the deviating firm is able to capture
fully the resulting storage demand rather than share it equally with the rival in the next period.
This is necessarily profitable if the discount factor is sufficiently high. As a result, the demandshift incentive leads to an equilibrium in which buyers are induced to store the good initially
at a relatively low price. The price in period 2 must then be higher, consistent with equilibrium
storage, in inverse proportion to the discount factor (the price difference will also reflect any
physical depreciation or spoilage of the good as well as any storage costs incurred by buyers). On
the other hand, if the discount factor is very low, then buyers require a large discount off of the
expected future price in order to be induced to store. The associated loss in revenue from selling to
the period-1 consumption demand at this discounted price makes it unprofitable for a firm to shift
demand. Interestingly, for intermediate values of the discount factor, both storage and no-storage
equilibria exist.
The welfare consequences of storage in our analysis are straightforward. Consumers benefit
unambiguously because prices are lower and consumption is higher on a period-by-period basis.
For firms, profits are lower because the demand-shift incentive leads to aggressive period-1
competition for storage demand and the firms would earn higher overall profits in the absence of
storage (a repetition of the static Cournot outcome). It is also clear that a dynamic inefficiency is
present, since production costs are incurred prior to consumption for the amount that is stored.
Imperfect competition (oligopoly) is crucial for our results on market dynamics. Under
perfect competition the market would remain at a price equal to marginal cost. Similarly, a
monopolist would charge the static monopoly price in each period, since inducing storage would
involve shifting future demand back to the current period where the price is lower. The absence of
a monopoly incentive highlights the role of market structure in our explanation, and potentially
constitutes a testable prediction. A second set of predictions involves countercyclical prices.
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First, on the endogenous side, Figure 1 illustrates that prices are low and quantities are high in
period 1, where storage demand shifts out from consumption demand, while prices are high and
quantities are low in period 2, where residual-demand shifts in from consumption demand. Second,
exogenous contractions in consumption demand still involve equilibrium storage outcomes and,
hence, rising prices.3
Demand shifting also occurs with durable goods in oligopoly market settings, where firms
have an incentive to sell rather than rent the good (Bucovetsky and Chilton, 1986; Bulow, 1986;
Carlton and Gertner, 1989). Equilibrium in these models, however, differs from that in our storablegood model. Consider, for example, the two-period rental and sale model in Bucovetsky and
Chilton (1986). Aggregate output (rentals plus sales) and the rental rate for period 1 always lie
on the static consumption demand curve; with a durable good, arbitrage involves only the sale
price relative to the present discounted value of rental prices. Demand shifting is then costless
in the sense that the rental rate in period 1 is not reduced when a unit is sold rather than rented.
In contrast, demand for a storable good exhibits a kink, departing from static demand, and this
reflects a storage arbitrage condition on the price of the good across periods (recall Figure 1). This
makes demand shifting (inducing storage) costly for a seller in a storable-goods market because
it reduces the period-1 price. While the durable-goods market has a positive level of sales for any
positive discount factor, we find that a storable-goods market will have a no-storage equilibrium
as well as a storage one at intermediate discount factors. The contrasting demand structures thus
create different strategic incentives and result in qualitatively different market outcomes (for
common underlying consumption demands).
Demand shifting also arises when sellers can write forward contracts with buyers. Allaz and
Vila (1993) show that if each of two sellers can forward contract with buyers arbitrarily many
times before the consumption date, then the equilibrium price must converge to marginal cost.
Intuitively, buyers with perfect foresight will not buy forward unless the contractual price is no
higher than the expected spot market price. As long as there is a residual demand above the marginal
cost, the sellers will find it profitable to contract to sell more. However, this equilibrium outcome
differs from that in our storable-good model. A key difference between forward contracting and
storage is that the former does not involve purchase in advance—all exchanges take place at the
date of consumption. Thus, buyers require no discount from the expected spot market price to
enter into a forward contract. In this sense, demand shifting via forward contracting is costless
(as in a durable-good market with rentals and sales) and, in equilibrium, there will always be a
positive level of forward contracting.4
The recent contribution of Hong, McAfee, and Nayyar (2002, HMN hereafter) also examines
buyer inventories and imperfect competition in a theoretical framework.5 In their model, all buyers
have the same valuation for the good and the focus is on heterogeneity with respect to price
searching and storage abilities. We focus on the heterogeneity of buyers with respect to valuation
for the good and, hence, variations in the extent of storage; all buyers in our model are able
to store the good. HMN model consumer storage and intratemporal price dispersion where, as
in Varian (1980), each oligopolist owns a captive monopoly market (captives) while competing
to attract additional buyers (shoppers) who purchase only from the lowest-priced firm and can
purchase for storage. Equilibrium involves price dispersion (mixed strategies). Mean prices exhibit
intertemporal cyclicality, depending on whether shoppers purchase for inventory at a low realized
price in the previous period, a property we also find for the price path in our two-period model.
3 Demand-accumulation models of durable goods (e.g., Sobel, 1991) also provide an explanation for price dynamics and intertemporal demand movements. Empirical work seeking to uncover evidence of such pricing, based on a
model of imperfect competition, is provided by Pesendorfer (2002).
4 Distinct from the idea of demand shifting is that of demand building, which arises in models of dynamic
competition with customer switching costs (see Beggs and Klemperer, 1992) or network externalities (see Katz and
Shapiro, 1985) where an increase in current market share improves future profits.
5 Trade deals and reduced wholesale prices are examined in the marketing literature by Lal, Little, and VillasBoas (1997). They find a mixed-strategy equilibrium for wholesale prices in which the retailer can forward buy and hold
inventory for later sale to brand-loyal customers. See also Jeuland and Narasimhan (1985).
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The fundamental economic assumptions, however, differ significantly across the two models.6 As
a result, the equilibrium forces driving several critical dimensions, such as the level of storage, the
effect of storage on future prices, and the existence of arbitrage gains to storage, reflect different
economic forces and result in differences in the outcomes. The contrast in model structures is
helpful for understanding the economic forces at work, and we discuss these points in more detail
when we study storage demand.
We describe the oligopoly storage model in Section 2. We then examine equilibrium storage
demand and price arbitrage in Section 3. In Section 4 we prove existence and then characterize
equilibria. In Section 5 we derive the monopoly outcome, and in Section 6 we provide the
equilibrium foundation for our residual-demand (rationing) approach. Section 7 concludes, and
all proofs are in the Appendix.
2. The model
Consider the following two-period game, with time indexed by t = 1, 2. Two firms, i and j,
each choose a quantity qti and qtj , respectively, to produce in period t. Let pt denote the price of
the good and qt = qti + qtj the total output in t. Each firm has an identical constant marginal cost
of production c, where c < 1. Both firms, as well as all buyers, discount future payoffs at δ < 1
per period.
Buyers act as price takers, and each buyer consumes either one unit of the good or zero per
period. The set of buyers seeking to consume in t = 1 is given by a continuum of total mass 1,
and buyer valuations for consumption at t = 1 are distributed as F(v). Thus, a buyer with value
v will purchase for consumption in t = 1 whenever v ≥ p1 , and hence consumption demand for
period 1 is
Q C1 ( p1 ) = 1 − F( p1 ).
(1)
We assume that F has support [0, 1] with F(0) = 0, F(1) = 1, and that F is twice continuously
differentiable with a strictly positive density f on [0, 1]. To ensure that marginal revenue is strictly
decreasing, we make the standard assumption that the inverse hazard rate, h(v) ≡ [1− F(v)]/ f (v),
is strictly decreasing in v.
Demand for consumption in t = 2 is similar, with the exception that we allow for an expansion
or contraction in overall market demand, via a demand parameter λ, in order to examine the impact
of market growth on equilibrium. When λ = 1, consumption demand is time invariant (constant
across t = 1, 2). λ > 1 corresponds to a growing market (outward demand shift) and λ < 1 to a
shrinking market (inward shift). To avoid trivial cases (a degenerate market at t = 2), we assume
λ > c.
Buyers seeking to consume in t = 2 form a continuum of total mass λ, where λF(v/λ)
buyers have a valuation for period-2 consumption that is less than or equal to v, where v ∈ [0, λ].
In the absence of storage opportunities, a buyer with value v will purchase for consumption in
t = 2 whenever v ≥ p2 and period-2 consumption demand will be λ[1 − F( p2 /λ)]. Note that
λ[1 − F( p2 /λ)]≷1 − F( p2 ) as λ≷1 so that λ corresponds unambiguously to an outward or inward
shift of consumption demand relative to period 1. For example, when F is the uniform distribution,
λ corresponds to a shift in the intercept of the (linear) demand function, since λ[1 − F( p2 /λ)] =
λ − p2 .
The model admits several interpretations for demand growth and the set of buyers. For the
benchmark case of λ = 1, simply imagine that all buyers participate in both of the t = 1, 2 markets
and that an individual buyer with values v1 and v2 for t = 1, 2 consumption has preferences
v1 − p1 + δ(v2 − p2 ). Alternatively, λ = 1 corresponds to each buyer having a time-invariant
6 In many respects, the models are complementary (beyond clear differences such as price versus quantity setting).
For example, one could introduce captive buyers into our model. Similarly, one could introduce valuation heterogeneity
in HMN. On the empirical side, both forms of buyer heterogeneity, valuation and storage/search, appear to be important;
see Hendel and Nevo (2005, forthcoming) and Erdem, Imai, and Keane (2003) for more on this point.
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valuation, v1 = v2 = v , where v ∼ F. To interpret demand growth (λ > 1), set v1 = v and
v2 = λv for these buyers and suppose that an additional mass λ − 1 of buyers, with v ∼ (λ − 1)F,
only value period-2 consumption (i.e., the additional buyers have v1 = 0 and v2 = λv ). For
a demand contraction, suppose that a fraction 1 − λ of the original buyers exit the market after
period 1 (equivalently, this fraction has v2 = 0). Alternatively, we can view the buyers for period-1
and period-2 consumption as distinct populations for which period-1(2) buyers have no value for
period-2(1) consumption. Finally, as noted below, the demand structure can also be generated
with consumers who have concave utility over continuous quantities and additively separable
two-period preferences.
Now consider storage. We make the critical assumption that buyers have the option of
purchasing the good in period 1 and storing it for consumption in period 2. Alternatively, a buyer
can wait and purchase in period 2. Let x ≥ 0 denote the total amount of the good stored by all
buyers in period 1. Period-2 demand must then account for the fact that any buyer who stored the
good will not have a consumption demand in period 2. The standard modelling construction for
residual demand in period 2 is to assume that x was purchased by buyers with values at the high
end of the distribution. We adopt this residual-demand approach, since it simplifies the analysis
relative to working with a more general rationing rule. We show in Section 6 that this is without
loss of generality and that it is the definition of equilibrium rather than the rationing rule that
determines the extent of storage.
Thus, we assume that storage x is held by buyers with high values for period-2 consumption.
Consequently, consumption demand in period 2 is from buyers with v ≤ λF −1 (1 − x/λ) and we
have
Q C2 ( p2 , x) = λ − x − λF( p2 /λ),
(2)
whereby inverse-consumption demand for period 2 is given by
P2 (q2 + x) = λF −1 (1 − (q2 + x)/λ).
(3)
The timing of events is as follows. At each t, the two firms simultaneously choose that
period’s output level. The current price is then determined by market clearing: aggregate demand
from buyers must equal total supply from the firms. To avoid information issues, suppose that
each firm observes the period-1 output of the other firm before producing for period 2; thus, we
have a proper subgame at the start of period 2.
Period-2 outcomes. The (subgame-perfect) equilibrium outcome in period 2, given storage
x ≥ 0, follows from standard Cournot analysis for the residual-demand curve (3). We record
these results now in order to streamline the subsequent analysis. In period 2, firm i chooses q2i to
maximize
π2 (q2i , q2j + x) ≡ P2 q2i + q2j + x − c q2i .
(4)
Then, a symmetric interior solution for i (and similarly for j) requires
q2i∗ P2 (q2∗ + x) + P2 (q2∗ + x) = c,
(5)
where q2i∗ = q2∗ /2. Define a threshold level of storage by x̄ = λ[1 − F(c/λ)]. We have the
following.
Lemma 1. For each x ≥ 0, there exists a unique equilibrium outcome for period 2 and the
equilibrium is symmetric across firms with q2i∗ = q2j∗ . For x < x̄, each firm produces a positive
output and price exceeds marginal cost. For x̄ ≤ x ≤ λ, each firm produces a quantity of zero.
The lemma follows from standard techniques for Cournot analysis (see Vives, 1999). The
inverse-hazard assumption implies that the marginal revenue of firm i is strictly decreasing in
i’s output, and this implies existence. Uniqueness follows from the best reply of each firm being
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downward sloping with a slope greater than −1 (this follows from a standard monotone comparative statics argument). Symmetry follows from the symmetry of best replies across players.
Period-2 output, q2∗ (x), is given by (5) upon noting that q2i∗ = q2∗ /2. We then have P2∗ (x) ≡
∗
P2 (q2 (x) + x) for the equilibrium period-2 price as a function of the storage level x. Each firm
earns 2 (x) = [P2∗ (x) − c]q2∗ (x)/2. Of course, when storage is sufficiently large (above x̄), no
buyer with a valuation above marginal cost is in the market in period 2. In this event, any price
above λF −1 (1− x/λ) will clear the market, since the resulting demand of zero balances the choice
of zero output by the firms. Whenever the period-2 market is active, we have the following.
Lemma 2. Suppose x < x̄. Then P2∗ (x) is differentiable in x, with P2∗ (x) < 0.
Thus, greater storage by buyers in period 1 results in a lower equilibrium price for period
2. We also note, for later reference, that P2∗ (x) increases with λ and, intuitively, an exogenous
increase in period-2 consumption demand must result in a higher price (see the proof of Lemma
2). With period-2 outcomes characterized relative to storage, we are ready to specify storage
decisions.
Storability and the buyer’s problem. Aggregate demand for period 1 must be derived
from the underlying purchase decisions of individual buyers in response to the current market
price and the expected future price. Letting p2e denote the buyers’ (common) expectation of the
period-2 market price, consider a buyer with value v ∈ [0, λ] for period-2 consumption and the
storage decision. Purchasing at t = 1 for p1 and then consuming at t = 2 has payoff δv − p1 .
By waiting until t = 2, the buyer can purchase at p2e for a payoff of δ(v − p2e ). Thus, whenever
p1 < δp2e the buyer strictly prefers storage to waiting. Of course, v must satisfy δv − p1 ≥ 0 for
the buyer to be willing to make the storage purchase. Thus, the storage decision rule for a buyer
with value v is given by

if p1 < δp2e and δv ≥ p1
1
e
D S ( p1 , p2 , v) = {0, 1} if p1 = δp2e and δv ≥ p1
0
if p1 > δp2e .
(6)
When p1 > δp2e , the current price is greater than the discounted expected future price, and
storage is strictly dominated. On the other hand, when p1 is below δp2e , all buyers with v > p1 /δ
strictly prefer to store the good; implicitly, no buyer expects to purchase the good in period 2
in this case. When equality holds, and p1 = δp2e , buyers with v > p1 /δ are indifferent between
storing the good and waiting to purchase. Effectively, discounting is a simple device for capturing
an economic “cost” to storage, as, for example, buyers will postpone purchases whenever prices
are expected to remain constant (or rise only a small amount relative to δ).
The storage decision rule, (6), also emerges in a continuous choice framework. Suppose,
for instance, that a large number of identical buyers each have a concave utility function over
consumption and can purchase any desired quantities. Then, the storage versus future consumption
decision rule follows the same price comparison as above. Further, if we interpret F in terms of
the (inverse) marginal utility of consumption, then aggregate demand will also coincide.7 We note
in passing that the model can accommodate a direct (additive) storage cost: if a buyer incurs cost
s to store the good for one period, then p1 + s < δp2e becomes the threshold for storage. Also, we
can allow for a spoilage rate over time: if 1 unit in period 2 requires the storage of 1/θ units, then
θ is equivalent to discounting from the buyer’s point of view.
We emphasize that aggregate storage demand, denoted by Q S ( p1 ), is endogenous in this
model. Buyers respond to current and expected future prices when making their storage decision.
7 Suppose the typical buyer chooses current consumption and storage and plans future consumption in response
to p1 and p2e to maximize U (c1 ) − p1 [c1 + d] + δ[U (c2 + d) − p2e c2 ]. Then, only one of d and c2 is positive at an
optimum (unless p1 = δp2e , in which case only the sum of d and c2 is determined) and the choice depends on p1 versus
δp2e . Further, since the optimal choices are characterized by U (c1∗ ) = p1 and U (d ∗ + c2∗ ) = min{ p1 /δ, p2e }, the same
aggregate consumption and storage demands emerge when we view F in terms of the inverse marginal utility.
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To be rational, the period-2 price expectations must be linked to how period-2 market outcomes
are affected by the effect of storage on future demand. We will derive Q S ( p1 ) once we have
introduced the definition of equilibrium.
Firms’ profit-maximization problem. Each firm maximizes the sum of current and future
discounted profits. In doing so, they must recognize that greater aggregate sales today imply a
smaller size for the residual market tomorrow. By encouraging buyers to store the good (producing
more today to drive prices lower), however, an individual firm can capture future market share
from its opponent. In this sense, higher individual sales today generate a greater share of the
intertemporal market.
In period 1, firm i chooses output q1i to maximize
max[P1 (q1i + q1j ) − c]q1i + δ2 (x),
q1i
(7)
for aggregate inverse demand P1 (q1 ) and x = Q S (P1 (q1i +q1j )). Similarly, firm j seeks to maximize
its total profit.
Definition of equilibrium. We can now define equilibrium, taking care to specify the
requirements on expectation formation for buyers, as well as period-1 storage demand and market
clearing. Since the period-2 outcomes are fully described above for any amount of period-1
storage, x, we omit their explicit specification from the equilibrium requirements.
Definition. A subgame-perfect rational-expectations equilibrium (henceforth, an equilibrium)
consists of period-2 price expectations, P2e ( p1 ), storage demand, Q S ( p1 ), inverse demand P1 (q1 ),
and outputs {q1i∗ , q1j∗ }, together with the associated period-2 outcomes, such that
(i) prices are market clearing: for any q1 ∈ [0, 1 + λ], at p1 = P1 (q1 ) we have Q C1 ( p1 ) +
Q S ( p1 ) = q 1 ;
(ii) buyer storage choices are optimal: Q S ( p1 ) aggregates individual buyers’ storage decisions, as described in (6), at ( p1 , P2e ( p1 )) for all p1 ∈ [0, 1];
(iii) buyer expectations are rational: P2e ( p1 ) = P2∗ (x) at x = Q S ( p1 ) for all p1 ∈ [0, 1];
(iv) firm profit maximization: given q1j∗ , q1i∗ solves the profit-maximization problem in (7),
and similarly for j.
Condition (i) states that the price in period 1 is set to clear the market in the sense that
if firms produce q1 in total, then buyers are willing to purchase this quantity at the price p1 .
Thus, each firm can employ P1 (q1 ) to assess the profit consequences of different output levels.
Condition (ii) specifies that whenever there are gains to buying in advance of consumption, buyers
will act to realize such gains. This storage choice depends critically on buyer price expectations,
and condition (iii) requires that buyers have rational beliefs about how future price depends on
current price. It is important to note that this condition applies not only for a realized equilibrium
price p1∗ but also at alternative period-1 prices that are off of the equilibrium path. Thus, if a firm
overproduces relative to equilibrium, then buyers accurately assess the implications for future
prices.8 Finally, condition (iv) is a standard duopoly (Nash) equilibrium requirement relative to
the payoff structure (which incorporates period-2 outcomes).
3. Storage demand and price arbitrage
In this section we characterize aggregate storage demand and the resulting restrictions on
prices. Given a price p1 , storage demand is endogenous because storage will affect the equilibrium
8 Buyer price expecations can equivalently be described directly in terms of P ∗ (x), if buyers observe the storage
2
level. The equilibrium definition implies that buyers can make the correct inference observing only the period-1 price.
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price in period 2. The resulting price in period 2 must, however, be consistent with the initial choice
of buyers to store the good in period 1. We begin with arbitrage restrictions on equilibrium prices.
Price arbitrage. We know, from (6), that buyers are indifferent about buying in advance
when p1 = δp2e . Given this, quantity is indeterminate and optimal storage requires only that x
lies between 0 and λ[1 − F( p1 /λδ)]. Equilibrium, however, implies a tighter characterization of
storage behavior.
Lemma 3. In equilibrium, if x > 0, then p1 = δ P2e ( p1 ); further, x < x̄ and q2∗ (x) > 0 also hold.
Intuitively, Lemma 3 states that whenever buyers choose to store the good in equilibrium,
prices leave them indifferent. Effectively, the option to store brings the discounted period-2 price
into equality with the current price and eliminates any strictly positive arbitrage gains. The proof
of Lemma 3 is also instructive as it implies that, in equilibrium, it can never be the case that all
period-2 consumption is bought in advance in period 1. In other words, there must be positive
sales in period 2.
Storage demand. We can now derive aggregate storage and consumption demand for period
1. Formally, for any given price p1 , we must find the aggregate storage quantity Q S ( p1 ) that is
consistent with an optimal storage choice (6) when buyers hold rational expectations about the
consequences of storage for period-2 prices, as required by condition (iii) for equilibrium.
To begin, recall from above (just after Lemma 2) that stronger period-2 demand (larger λ)
will increase the period-2 price. Intuitively, we can expect that extremely large or small values
for λ will lead to a trivial storage outcome. To rule out such cases, which are easily dealt with as
limiting cases of the analysis, assume that
c < δp2C < 1,
where p2C ≡ P2∗ (0) is simply the (static) equilibrium Cournot price for period 2 (i.e., no storage
and demand of q = λ[1 − F( p/λ)] in period 2); implicitly, λ is in an open interval containing 1.
Now consider storage demand when p1 > δp2C . We claim buyers will not store at such a
price. The reason is that when x = 0, we will have a relatively low period-2 price. Formally,
P2∗ (0) = p2C = P2e ( p1 ), and we then have p1 > δ P2e ( p1 ), which means that waiting to purchase in
period 2 is optimal. Any x > 0 would only depress the period-2 price and reinforce the decision
to wait. Hence, we have Q S ( p1 ) = 0 at any such p1 .
Now suppose p1 < δp2C . Then, if x = 0, we have P2∗ (0) = p2C = P2e ( p1 ). Hence, p1 <
e
δ P2 ( p1 ), and so all buyers with a period-2 valuation above p1 /δ have a strict incentive to purchase
in period 1 and store the good. Consequently, Q S ( p1 ) > 0 at any such price. In equilibrium, the
extent of storage must be sufficiently large to pull the period-2 price down to p1 /δ. To find this
level of storage, recall that by no-arbitrage (Lemma 3), we have p1 = δ P2∗ (x) whenever x > 0.
Then, in order for the aggregate storage behavior of buyers to be consistent with equilibrium
expectations for period 2, we must have
Q S ( p1 ) ≡ x = [P2∗ ]−1 ( p1 /δ).
(8)
Recall that P2∗ (x) is well defined and invertible (as it is strictly decreasing) for all x ≤ x̄.
If x > x̄ (i.e., when p1 < δc), then residual period-2 demand is such that no firm can profitably
produce and sell in period 2. As noted after Lemma 1, this market outcome with no sales is
consistent with any period-2 price that is sufficiently high. To have Q S ( p1 ) well defined for all p1 ,
it is convenient to adopt the convention that if p1 < δc, then [P2∗ ]−1 ( p1 /δ) ≡ λ[1 − F( p1 /λδ)].
Intuitively, if p1 < δc, then all period-2 buyers who value the good at or above p1 /δ will purchase
in period 1, knowing that in period 2 there will be no market transactions. We have thus established
the following.
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Lemma 4. In equilibrium, Q S ( p1 ) is nonincreasing and differentiable everywhere (except at
p1 = δp2C ) for all p1 ≥ 0. Further,

0
Q S ( p1 ) =

[P2∗ ]−1 ( p1 /δ)
if p1 > δp2C
(9)
if p1 ≤ δp2C .
Aggregate demand in period 1, which is the sum of consumption and storage demand, is
then

 1 − F( p1 )
Q 1 ( p1 ) =

1 − F( p1 ) + [P2∗ ]−1 ( p1 /δ)
if p1 > δp2C
(10)
if p1 ≤ δp2C .
Since F and [P2∗ ]−1 are each differentiable (except at p1 = δp2C ), the inverse aggregatedemand function, P1 (q1 ), is differentiable everywhere except at the point q K ≡ 1 − F(δp2C ).
Moreover, since P2∗ (x) is strictly decreasing, we see that inverse aggregate period-1 demand
decreases everywhere and has a kink at the point q K . Intuitively, at prices above δp2C , buyers do
not seek to buy in advance because the period-2 price will be sufficiently low (at p2C ). In this
case, aggregate period-1 demand consists only of consumption demand for period 1. At prices
below δp2C , however, some buyers purchase their period-2 consumption in advance and store the
good. Consequently, at any price below δp2C , the aggregate period-1 demand lies above the level
of period-1 consumption demand. Graphically, this corresponds to a kink (see Figure 1 above) at
the price δp2C and quantity q K where demand rotates outward, becoming “flatter” in the sense that
the slope is less negative, once buyers start augmenting their consumption demands with storage
demand.
Storage by buyers thus provides an equilibrium rationale for a kinked-demand function in
oligopoly markets. The economic logic for the kink, however, extends beyond the oligopoly
setting. As we will see in Section 5, the period-2 pricing behavior of a monopoly supplier also
generates a kink in period-1 demand, albeit at a higher price. In both of the oligopoly and monopoly
settings, period-2 prices are determined via the residual-demand structure given prior storage.
Thus, the kink in demand arises when current and future prices are related via the arbitrage
condition and the future price is negatively related to the extent of storage.
The contrast in storage demand between our model and that in HMN is helpful for understanding the economic forces created by the underlying source of buyer heterogeneity. In our
model, period-2 prices fall with the extent of storage (recall Lemma 2) because buyers have different valuations for the good. Thus, whenever prices induce storage, it is necessarily the case
that buyers with valuations that are high relative to current prices are the ones who store the
good. Buyers from the lower end of the valuation distribution necessarily remain in the market
and seek to purchase for consumption in period 2. In contrast, in HMN, heterogeneity takes the
form of captives versus shoppers. There, storage implies that shoppers are removed from future
consumption demand. Storage then leads to higher prices (a higher mean price in the equilibrium
dispersion) in the future period, since each firm has a stronger incentive to price high and extract
surplus from their captive buyers.
Two further points on heterogeneity now follow directly. First, with valuation heterogeneity,
the extent of storage can vary smoothly and it is part of the equilibrium determination. With
a common valuation, all buyers tend to be on the same side of the storage decision. Thus, the
storage level in HMN coincides with the number of shoppers when storage occurs, and it is only
the price dispersion that adjusts in equilibrium. A common property of both models, however,
is the presence of an externality among buyers: storage at the aggregate level, as discussed by
HMN, influences future prices. Second, valuation heterogeneity leads to a tight arbitrage linkage
in prices across periods when storage occurs (recall Lemma 3), for the same reason that the extent
of storage demand varies as buyers with different valuations become active.
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A linear-demand example. Suppose F is uniform on [0, 1]. Then (3) reduces to P2 (q2 +x) =
λ − (q2 + x). The corresponding period-2 price is easily verified to be P2∗ (x) = (1/3)[λ − x + 2c].
With p1 = δ P2∗ (x) from Lemma 3, we have, for 0 < x < (λ − c),
p1 = δ P2∗ (x) =
δ
[λ − x + 2c].
3
(11)
Upon rearranging, we have storage demand of
Q S ( p1 ) ≡ x = λ + 2c − 3 p1 /δ.
(12)
Aggregate demand in period 1 is then given by
1 − p
1
Q 1 ( p1 ) = 1 + λ + 2c − (1 + 3/δ) p1
1 + λ − (1 + 1/δ) p1
if p1 > δ(λ + 2c)/3
if δ(λ + 2c)/3 ≥ p1 ≥ δc
if δc > p1 .
(13)
The period-1 demand panel in Figure 1 exhibits the main qualitative features of this example.
4. Equilibria
The basic economic forces that will determine equilibrium outcomes can now be identified.
On the one hand, the static Cournot outcome is a natural candidate. Since this outcome lies above
the kink in the demand function, it is a local simultaneous best response for the firms: small
changes in quantity will not trigger storage, and each firm is at a local profit maximum. The kink
in demand, on the other hand, provides a market “carrot”: a sufficiently large output increase
will trigger storage, and the resulting increase in sales (above period-1 consumption demand)
may make the deviation profitable. We refer to this as the “demand-shift incentive,” and our
main result, established below, is that this incentive is sufficiently strong that equilibrium always
involves storage whenever δ is not too small.
Thus, our goal in this section is to establish the existence of equilibrium and characterize when
storage occurs. The results apply to a fairly broad set of demand settings (value distributions) and
only rely on the assumption of downward-sloping marginal revenue (the inverse-hazard property).
While examples, including the linear-demand example presented later in this section, have bestresponse functions with a simple structure, this cannot be expected in general. Consequently, this
section has a more prominent technical component.
Figure 2 provides an overview of the main equilibrium issues. The solid curve shows the
global best response of i to output q j from j. The first feature to note is how i responds to the
output level of q B by j. Based purely on static profit incentives (period-1 consumption demand), i
would respond to q B such that aggregate output is at the kink q K ; thus, point B lies on the familiar
static Cournot best response (labelled N ). But marginal revenue jumps upward at q K , so this
cannot be i’s global best response. Instead, the best response lies directly above at point C, on the
S curve, where aggregate output involves positive storage. The higher output for i reflects the
demand-shift incentive. Intuitively, higher output from j always moves the market closer to the
demand kink and increases the incentive for i to respond with an output that induces storage.
Point A in Figure 2 is where the demand-shift incentive for i first dominates the static
incentive. Thus, the best response jumps from point A, the static Cournot response, to the higher
level at point D. Since the jump occurs before N hits the 45◦ line, the static Cournot outcome is
not an equilibrium. Instead, the equilibrium is at point E in Figure 2, where S hits the 45◦ line.
In general, however, the S branch of the best response need not be as well behaved as in Figure
2. Further, the location of the jump point q A is critical for when equilibrium involves storage, and
we must relate it to the effects of demand growth (λ) and the implicit cost of storage (δ).
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FIGURE 2
BEST-RESPONSE AND STORAGE EQUILIBRIUM
Existence. To begin the equilibrium analysis, recall the discounted sum of profits over both
periods from (7):
(q i , q j ) ≡ [P1 (q i + q j ) − c]q i + δ2 (x),
(14)
where x = Q S (P1 (q i + q j )); we drop the t subscript when it is clear that we are considering
period-1 outputs. In period 1, each firm chooses quantity to maximize this discounted profit sum.
Given the kink at q K ≡ 1 − F(δp2C ), the best-response quantity choice will exhibit discontinuities
and may also be multivalued. Consequently, we work with the best-response correspondence
(q j ) = argmax (q i , q j ).
(15)
0≤q i ≤1+λ
Quantity choices above 1 + λ imply a price of zero in period 1. Since (q i , q j ) is continuous
on a compact set, we know that (q j ) is nonempty for all q j ∈ [0, 1 + λ]. We must identify when
a firm will find it optimal to induce buyers to store the good. To this end, we can examine the
best-response problem in terms of two subproblems. First, consider the best-response problem
when i is restricted to choosing a quantity that induces storage by buyers, as given by
S (q j ) =
argmax
max{q K
−q j ,0}≤q i ≤1+λ
(q i , q j ),
(16)
where q j ∈ [0, 1 + λ]. By construction, the domain for the q i choice implies that we are on the
storage region of the period-1-demand function. As with , we know S is nonempty. Next,
consider the best-response problem when i can feasibly choose a quantity that does not induce
storage, as given by
N (q j ) = argmax [F −1 (1 − (q i + q j )) − c]q i ,
(17)
0≤q i ≤q K −q j
where 0 ≤ q j ≤ q K . The specified quantity ranges imply that we are in the no-storage region
of period-1 demand. Consequently, the payoff (q i , q j ) reduces to the period-1 profit flow plus
δ2 (0). In fact, it is clear that N (q j ) must be a singleton set, since it is a constrained version of the
best-response problem for a firm that faces the (static) period-1 consumption-demand function.
This best-response function is (static period-1 Cournot market)
ϕC (q j ) = argmax [F −1 (1 − (q i + q j )) − c]q i .
(18)
0≤q i ≤1+λ
It is routine to verify (a special case of Lemma 1) that ϕC (q j ) = 0 when q j ≥ 1 − F(c).
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Further, when q j < 1 − F(c), we find ϕC (q j ) = 1 − F( p) − q j , at the unique p that satisfies
p−c = [1− F( p)−q j ]/ f ( p); in this case, ϕC (q j ) is positive, differentiable, and strictly decreasing
with a slope greater than −1 (under the hazard assumption on F).
We employ these best responses to identify when storage arises. For q j > q K , the opposing
firm is at a high output level and we have (q j ) = S (q j ), since storage necessarily occurs at
these output levels. On the other hand, when q j ≤ q K , and the opposing firm is producing at a
relatively low level, we have (q j ) ⊂ S (q j ) ∪ N (q j ). Intuitively, if the period-1 output of
firm j is relatively low, i can restrict output so as not to induce storage. Each firm then earns the
per-firm Cournot profit in period 2 corresponding to the full consumption demand. Alternatively,
i can expand output to induce storage by buyers, reducing the period-1 price as well as the
per-firm period-2 profits (due to the reduced consumption demand in period 2). However, in
expanding output, i alone sells to the entire shifted period-2 demand, which if not shifted to
period 1 would have been shared equally between i and j in period 2. As a result, the higher share
of the intertemporal output that i captures in period 1 (at the expense of j’s period-2 sales) may
make it profitable for i to expand output and induce storage.
The next lemma relates the best responses and shows that an equilibrium exists.
Lemma 5. The following properties hold for the best-response correspondence:
(i) (q j ) ≡ {q i + q j | q i ∈ (q j ), q j ∈ [0, 1 + λ]} is a strongly increasing correspondence;
(ii) if N (q̂ j ) ⊂ (q̂ j ) for some q̂ j , then (q j ) = N (q j ) for all q j < q̂ j ; further, if
q̂ i + q̂ j < q K , where q̂ i ∈ N (q̂ j ), then S (q j ) ∩ (q j ) = ∅;
(iii) if S (q̂ j ) ⊂ (q̂ j ) for some q̂ j , then (q j ) = S (q j ) for all q j > q̂ j ; further, if
q̂ i + q̂ j > q K , where q̂ i ∈ S (q̂ j ), then N (q j ) ∩ (q j ) = ∅.
Finally, there exists an equilibrium.
In a static Cournot game, the aggregate quantity analog of is strongly increasing whenever aggregate inverse demand is strictly decreasing (see Vives, 1999). Even though our model
is dynamic, continues to be strongly increasing because equilibrium period-2 profit is fully
determined by the aggregate period-1 output, q1 , via x. Existence of (pure-strategy) equilibrium
follows from the fact that reaction functions do not have discontinuities of the “jump down”
variety, and therefore must intersect the 45◦ line (Tarski’s intersection point theorem; see also
Roberts and Sonnenschein (1976)).
Storage equilibria. When is the demand-shift incentive sufficiently strong that a firm will
induce storage? Of course, this is trivial if q j is sufficiently large (above q K ). The important case
arises when q j is relatively low and firm i has the option to produce at a low level where storage
is not induced, as well as the option to produce at a high level and induce storage. We find that it
is typically optimal to induce storage well before j’s output level reaches q K . To this end, let q B
denote the unique output for j at which ϕC (q B ) + q B ≡ q K . That is, q B induces a static Cournot
best response from i such that we are at the kink in the period-1 demand function. Recalling that
q K ≡ 1 − F(δp2C ) ∈ [0, 1], we see that q B = q K whenever q K ≥ 1 − F(c) or, equivalently, when
δp2C ≤ c. Otherwise, we have 0 < q B < q K < 1 − F(c).
Now consider the first time can jump from inducing no storage to storage. At q j = q B ,
we know from above that ϕC (q B ) + q B = q K . Hence, a global best response for i to q j = q B will
necessarily induce storage, since the upward jump in marginal revenue for i when total output
is at the kink implies that i can increase profit by choosing q i > ϕC (q B ). By Lemma 5 (iii), we
then have (q j ) = S (q j ) for all q j ≥ q B , so the first time the best response for i jumps from no
storage to storage must occur at or below q j = q B . Defining q A ≡ inf{q j | S (q j ) ⊂ (q j )},
we have q A ≤ q B , and, by Lemma 5 (ii), q A is the first and only time the best response jumps
from no storage to storage. We then have the following.
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Lemma 6. Let α ≡ 1 + (1/2)q2∗ (0). Then q A < q B if (DS) holds, where (DS) is the inequality
δ>
c
.
αc + (1 − α) p2C
Recall that the period-1 demand function has a kink at the price δp2C and quantity q K and
“flattens out” at lower prices where storage demand is positive. The kink implies that there is a
jump in marginal revenue at q K . Lemma 6 establishes a threshold (DS) for demand storage. Since
the demand function “flattens” at q K , there is a jump in marginal revenue at q K , implying that,
purely from the revenue standpoint, i is better off increasing its own output. (Thus, aggregate
output can never be equal to q K in equilibrium.) This gain is tempered by the reduced period-2
profit for i as a result of the consequent reduction in residual demand. Therefore, the net effect
favors demand shifting only if the jump in marginal revenue is sufficiently large. To gain intuition
for (DS), consider the special case of λ = 1 (identical consumption demand in both periods). The
kink in demand is located at price δp2C and quantity q K = 1 − F(δp2C ). But with λ = 1, the static
Cournot equilibrium outcomes for periods 1 and 2 coincide at p2C . Thus, as δ increases, the kink
in period-1 demand is moving closer to the static Cournot outcome for period 1 (reaching it in
the limit when δ = 1). Similarly, q B is approaching the per-firm static Cournot output. Thus, as δ
becomes large, if j produces at q B , then a very small increase in output by i above the Cournot
level will push the period-1 outcome beyond the demand kink, and, as a result, firm i will benefit
from the associated upward jump in marginal revenue. Of course, this logic continues to apply
if q j is sufficiently close to q B , and, therefore, the best response of i will jump from no storage
to storage before j’s output reaches q B (i.e., q A < q B must hold). Thus, we see that (DS) is a
sufficient condition on δ and, implicitly, λ such that q A is strictly below q B . Since p2C > c, it
follows directly that 1 > c/[αc + (1 − α) p2C ], thereby establishing the validity of the threshold
for δ in (DS).
We then have the following.
Proposition 1. There exist δ ∗ < 1 and λ∗ < 1 such that, for any discount factor δ > δ ∗ and any
demand growth parameter λ > λ∗ , all equilibria have strictly positive storage.
With λ = 1 and δ sufficiently large, equilibrium storage implies that price increases from
periods 1 to 2 even though the exogenous periods-1 and -2 consumption demands are identical.
More subtly, we see that the equilibrium price increases even though the endogenous period-2
market demand, reflecting positive storage, is less than the period-1 market demand (in equilibrium, quantity demanded in period 2, at any given price, is strictly less than that for period 1).
Of course, a storage equilibrium will continue to exhibit an increasing price path when λ > 1
(consumption demand is growing). Note, however, that storage equilibrium prices are still below
the prices corresponding to the static Cournot equilibrium outcomes for each period (this always
holds for period 2 and will also hold for period 1 except when λ is very large). A more surprising
property is the prediction that prices will increase even in market settings where λ < 1 and
consumption demand is declining. Formally, for δ > δ ∗ , there exist values of λ strictly less than
one, such that any equilibrium must involve strictly positive storage and, therefore, an increasing
price path. In other words, an exogenous decline in consumption demand is accompanied by an
increase in the market price.9
No-storage and multiple equilibria. We also want to examine how demand growth and
discounting influence the set of equilibria. The key to this involves understanding how q A , the
9 A two-period setting is limited with respect to the dynamics of exogenous shifts in consumption demand in that
only one of the market responses (in anticipation of the shift or subsequent to the shift) can be examined. The recent
empirical work by Chevalier, Kashyap, and Rossi (2003) examines pricing with respect to imperfect competition and finds
countercyclical prices during demand peaks. To the extent that normal periods of demand correspond to no-storage (static)
outcomes, a shift to storage outcomes at peak demand times is then consistent with lower prices and higher quantities.
Clearly, however, a multiperiod model is needed to address these dynamic transitions.
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best-response jump point, moves as δ and λ vary. We will focus on δ; to this end, let us fix λ = 1
for simplicity.10 We then have the following.
Lemma 7. Let λ = 1. Then, for δ ∈ [0, 1], the best-response jump point q A is continuous and
nonincreasing in δ. Further, q A is strictly decreasing whenever q A > 0.
Lemmas 6 and 7 share a common intuition. As δ increases, storage demand shifts up and the
kink occurs at a higher price. Thus, holding q j fixed, it becomes more profitable for i to increase
output. Since period-1 consumption demand is independent of δ, this increase in i’s output makes
equilibrium storage more likely. As a result, whenever positive, q A decreases with an increase in
δ. We then have the following.
Proposition 2. Let λ = 1. Then there exists a discount factor δ n with c/ p1C < δ n < 1 such that
the equilibrium with no storage exists if and only if δ ≤ δ n .
Propositions 1 and 2 together with equilibrium existence (Lemma 5) imply that for some
values of δ strictly less than δ n , there may exist both storage and no-storage equilibria. Recall
that the no-storage equilibrium is simply the static Cournot equilibrium. Further, since storage
is induced only at period-1 prices that are lower than the discounted static Cournot price from
period 2, this implies that from the seller’s perspective, the no-storage equilibrium is Pareto
superior to any storage equilibrium. Interest in such multiplicity of equilibria is then natural. As
a comparative static exercise, we wish to identify the set of δ values for which there exist both
no-storage and storage equilibria. Toward that end, observe that storage equilibria correspond to
points of intersection between the 45◦ line and the correspondence ∩ S . Naturally, in order
to identify the full set of values of δ for which storage equilibria exist, we need to know how S
behaves as we vary δ. It turns out that, without imposing stronger assumptions on demand, little
can be said about whether or not S is monotonic with respect to δ. However, the case in which
F is uniform has a closed-form solution.
Proposition 3. Suppose F is uniform and λ = 1. Then there exist discount factors δ s and δ n with
c/ p1C < δ s < δ n < 1 such that
(i) if δ n < δ, then there exists a unique equilibrium and it has strictly positive storage;
(ii) if δ s ≤ δ ≤ δ n , then there exist exactly two equilibria, one with no storage (the repeated
static Cournot outcome) and one with strictly positive storage;
(iii) if δ < δ s , then there exists a unique equilibrium and it has no storage (the repeated
static Cournot outcome).
Recall that 1/δ serves as the storage cost to buyers. When δ is sufficiently high, buyers require
a small discount off the expected future price in order to be induced to store. Then, inducing storage
is relatively inexpensive to firms, and each firm, in an effort to shift future demand from its rival,
overproduces in period 1 (relative to its static Cournot output), resulting in buyer storage. When
δ is sufficiently low, buyers require a large discount in order to store. As a result, each firm finds
it too costly to capture future market share from its rival and, in equilibrium, each produces its
static Cournot output so that no storage occurs.
Finally, for storage costs in the intermediate range, there are two equilibria. One (static
Cournot) has no storage. In this case, the rival firm is producing a relatively small output, and
inducing storage involves such a large price reduction that a firm finds it too costly to do so.
The other equilibrium (possibly multiple when F is not uniform) is one in which both firms
overproduce, thus inducing storage. In this case, the rival firm is producing a relatively high
output—as a result, the kink in the demand function is sufficiently close that it is profitable for a
firm to overproduce and induce storage.
10 The argument that follows can be applied at λ above and below one. The added complication is that we must
also introduce corner conditions on the critical discount factors, since more extreme λ values will necessarily eliminate
storage or no-storage equilibria across all discount factors. Even with λ = 1, the proof of Lemma 7 is fairly involved.
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With regard to our demand kink, one might expect buyer heterogeneity in storage costs to
“smooth out” the kink. However, for a simple example in which buyers differ in the discount (or
spoilage) factor and δ ∼ uniform [0, δ], the kink persists but the location moves to δ p2C . Thus,
when δ = 1 = λ, the demand kink coincides with the Cournot price and only storage equilibria
remain. (We thank the Editor for suggesting that we examine this point.)
A numerical example. Extending our linear-demand example, consider λ = 1 and c = 0.
From Proposition 3, a no-storage (i.e., Cournot) outcome and a storage outcome are the two
candidates for equilibria. Thus, we solve explicitly to identify the critical discount factors, δ s and
δ n , that characterize existence. Solving for the no-storage outcome yields qci∗ = 1/3 (the Cournot
outcome). Next, from (16), the first-order condition for the storage best response of i is
2
3
2 − [q j + 2 S (q j )] − (1 − x)
= 0.
9
3+δ
Storage is, from (12), x ≡ Q S (P1 (q1 )) = [3q1 − (3 − δ)]/(3 + δ), and then we find S (q j ) =
[(1+δ)(2−q j )]/[2(2+δ)]. Solving for the symmetric outcome, we find that qsi∗ = 2(1+δ)/(5+3δ)
is the candidate equilibrium storage outcome.
We find the critical discount factors, δ s and δ n , by determining when each of qci∗ and qsi∗ is a
global best response and, hence, an equilibrium output. For the no-storage outcome, we employ
(5) to calculate the profit (qci∗ , qci∗ ) and compare this to the profit ( S (qci∗ ), qci∗ ) for an optimal
deviation by i that involves storage (taking care to note that the corner solution for S will apply at
small δ). Solving, we find δ n = .825. Similarly, for the storage outcome, we compare (qsi∗ , qsi∗ )
to the deviation profit ( N (qsi∗ ), qsi∗ ) and then solve to find δ s = .626. For a graph of these
outputs relative to period-1 demand, refer back to Figure 1.
5. Monopoly prices
Consider the dynamic pricing problem confronting a monopolist who faces buyers with
rational storage behavior. We are interested in settings where the monopolist cannot commit to
period-2 prices in advance. Then, given storage of x ≥ 0, the monopoly problem in period 2 is to
choose p2 to maximize
π2M ( p2 , x) ≡ ( p2 − c)Q C2 ( p2 , x),
with Q C2 ( p2 , x) given by (2). For large storage, x ≥ x̄, it is optimal to shut down in period 2.
When x < x̄, the monopolist chooses to price at the unique P2M (x) at which marginal revenue
equals marginal cost, namely, the solution to
p2 −
λ[1 − F( p2 /λ)] − x
= c.
f ( p2 /λ)
(These results follow directly from the proof of Lemma 1 for a rival firm output of zero.) Of course,
quantity then follows from demand, Q 2M (x) = Q C2 (P2M (x), x). The period-2 monopoly payoff is
then 2M (x) = (P2M (x) − c)Q 2M (x), and, by the envelope theorem, 2M (x) = −(P2M (x) − c).
Finally, note that we can obtain the static period-1 monopoly solution simply by setting x = 0
and λ = 1 in the above period-2 problem; for later reference, let P1M denote this monopoly price.
In full analogy to the duopoly case, we find that rational storage behavior by buyers implies
Q 1S ( p1 ) ≡ x =
0
if p1 > δ P2M (0)
M −1
[P ] ( p1 /δ) if p1 ≤ δ P2M (0),
(19)
where P2M (0) is the optimal monopoly price in period 2 when x = 0 (for the full-consumption
demand of P2 (q2 ) = λF −1 (1 − q2 /λ)). Also, as with Lemma 3, we find that x > 0 implies
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THE RAND JOURNAL OF ECONOMICS
p1 = δ P2M (x). We can now examine the period-1 problem for the monopolist in terms of two
components: (i) whether to induce storage or not and then (ii) what price to charge. Choosing no
storage (x = 0) constrains the price choice to be high, and we have the no-storage (NS) problem
max ( p1 − c)Q C1 ( p1 ) + δ2M (0).
p1 ≥δ P2M (0)
Choosing storage (x > 0) necessarily fixes p1 = δ P2M (x), and we have the storage (S)
problem
max[δ P2M (x) − c][Q C1 (δ P2M (x)) + x] + δ2M (x).
x>0
(The payoffs for the two problems coincide at x = 0 and p1 = δ P2M (0).) We then have the
following.
Proposition 4. Suppose that P1M > δ P2M (0). Then the monopolist maximizes profit by charging
the static monopoly price in period 1, p1∗ = P1M , and induces no storage in equilibrium.
On the equilibrium path the monopolist then charges P2M (0) in period 2. The condition on
static monopoly prices in Proposition 4 is a fairly weak requirement. In the benchmark case of
constant demand across periods, the condition holds for all discount rates, since P1M = P2M (0)
when λ = 1. It also holds for a range of λ above one when δ < 1, and it clearly holds for all
λ < 1.11
Proposition 4 is intuitive: a monopolist has no strategic incentive to try to shift demand. The
only potential motivation for demand shifting by the monopolist is discounting, as each dollar of
period-2 profit is worth less than a dollar of period-1 profit. However, if buyers have the same
discount factor as the monopolist, then rational storage requires that the period-1 price be exactly
equal to the discounted period-2 price. As a result, on the marginal unit that the monopolist sells
toward storage, the lower period-1 price exactly offsets the gain from demand shifting. On the
other hand, if P1M > δ P2M (0), then any pair of period-1 and -2 prices that induce buyer storage
must be less than the static (per-period) profit-maximizing monopoly prices. Therefore, any pair
of prices that induces buyer storage in a monopoly will result in a profit that is strictly less than
the profit from charging the optimal static monopoly prices.
6. Equilibrium demand rationing
We have assumed that the period-2 demand curve follows a standard residual-demand construction, with the storage quantity x always being held by buyers at the high end of the valuation
distribution. Different rationing rules, however, imply different period-2 outcomes. In this section we show that the definition of equilibrium, including perfect foresight, determines a unique
storage demand quantity that is independent of the rationing rule. The residual-demand rule can
then be regarded as a convenient simplifying assumption.
A rationing rule is a way of assigning willing buyers to storage. This situation arises only
when p1 = δp2e ; otherwise, storage demand is uniquely determined by (6). A general rationing
rule, R(v, p1 ), maps buyer valuations into {0, 1}. The interpretation is that when p1 = δp2e , a
buyer with valuation v obtains the good for storage if R(v, p1 ) = 1. Consistency requires that
R(v, p1 ) ≡ 0 for all v < p1 /δ, since these buyers strictly prefer not to store at p1 . For period-2
demand, let D0 (v; p1 ) ≡ {ṽ | p1 /δ ≤ ṽ ≤ v and R(ṽ, p1 ) = 0} denote the set of buyers with
valuation at or below v who were willing to store the good but did not obtain it under R. Since
these buyers remain in the market (set λ = 1 for simplicity), the buyer distribution for period 2 is
given by
F(v)
if 0 ≤ v < p1 /δ
G(v, p1 ) =
F( p1 /δ) + F[D0 (v, p1 )] if p1 /δ ≤ v ≤ 1,
11 Even when λ is sufficiently above one that P M < δ P M (0), a monopolist may still find it optimal to remain at
1
2
the “corner” and keep p1 = δ P2M (0) to avoid storage. This is true for all λ when F is uniform [0, 1].
© RAND 2005.
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ANTON AND DAS VARMA / 537
where F[D0 (v, p1 )] ≡ D0 (v, p1 ) d F denotes the measure of buyers in D0 (v, p1 ) under F. Since
F is atomless, we have F[D0 ( p1 /δ, p1 )] = 0 and G(v, p1 ) is continuous at v = p1 /δ. Storage is
then given by x = G(1, p1 ) − G( p1 /δ, p1 ), and period-2 demand under R is
Q 2R ( p2 , p1 ) = G(1, p1 ) − G( p2 , p1 ).
Note that p1 enters because rationing arises when p1 = δp2e and we need to consider period-2
demand in out-of-equilibrium situations where p2 = p2e .
Now consider how storage and equilibrium are related. The analog of Lemma 3 is the
following.
Lemma 8. Consider any rationing rule R. If x > 0, then p1 = δ P2e ( p1 ). For the monopoly market,
x > 0 if and only if p1 < δ P2M (0). For the duopoly market, x > 0 if and only if p1 < δp2C .
Thus, the particular rationing scheme is irrelevant to the (arbitrage) equilibrium-price relationship: whenever storage is positive, the period-1 price equals the discounted period-2 price. In
addition, equilibrium storage occurs in exactly the same price regions as before, namely, when
the period-1 price is below the discounted static equilibrium price.
The final step is to consider the exact level of storage that can occur in equilibrium. We have
the following.
Proposition 5. Consider any rationing rule R. If p1 ≤ δ P2M (0), then storage demand in equilibrium for the monopoly market is given by Q S ( p1 ) = [P2M ]−1 ( p1 /δ) and zero for higher p1 . For
the duopoly market, replace [P2M ]−1 with [P2∗ ]−1 .
Again, this is exactly what we found for the residual-demand rule. Thus, the equilibrium
requirements for storage by buyers and seller optimization are sufficient to pin down the level
of storage demand, independent of the particular rationing rule. It is important to realize that,
in equilibrium, positive storage implies that rationing is always occurring: when p1 = δ P2e ( p1 ),
all buyers with v ≥ δp1 are indifferent. All of these buyers, however, cannot store the good
in equilibrium, as the resulting lack of demand would lead to a period-2 price below p1 /δ. In
equilibrium the seller(s) must find it optimal to behave in period 2 such that a market price of
p2 = p1 /δ prevails. In turn, this implies that among the indifferent buyers, sufficiently many of the
relatively low-value ones must remain in the market. This rules out, for example, a proportional
or low-end rationing rule. The residual-demand rationing rule, however, is fully consistent with
this requirement.
7. Conclusion
The rising price path in the storage equilibrium of our two-period model raises the important
issue of long-run market dynamics. As we have argued, equilibrium storage implies rising prices,
which cannot continue indefinitely. Thus, equilibrium price and associated storage (inventory)
cycles become a distinct possibility. This is an important topic for future research on storable
goods.
Appendix
Proofs of Lemmas 1-3, 5-8 and Propositions 1-5 follow.
Proof of Lemma 1. To prove the lemma, it is enough to establish that the period-2 marginal revenue of each firm is
decreasing in its own output (see the discussion following Lemma 1 in the main text). With constant marginal cost, this
implies that each firm’s period-2 payoff function (symmetric across the firms) is quasi-concave in its own output. It then
follows that each firm has a unique optimal output, and, furthermore, these two outputs are equal in equilibrium.
Define the period-2 revenue of firm i by R(q2i , q2j + x) ≡ P2 (q2i + q2j + x)q2i . Note that the period-2 price falls to
zero when x j ≡ q2j + x ≥ λ and it falls below c when x j ≥ x̄. Fixing x j < λ, we have
M Ri ≡
© RAND 2005.
q2i
∂
R(q2i , x j ) = p −
,
f ( p/λ)
∂q2i
mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
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THE RAND JOURNAL OF ECONOMICS
for any q2i ≤ λ − x j , where the M Ri expression is evaluated at p = P2 (q2i + x j ). Differentiating again, we have
M Ri ≡
q i f ( p/λ)
∂
2
M Ri = −
.
− 2
i
f ( p/λ)
λ f ( p/λ)3
∂q2
Clearly, M Ri < 0 holds at q2i = 0. Taking q2i > 0 and noting that λ[1 − F( p/λ)] − x j = q2i , we have M Ri < 0 ⇐⇒
2λ
f ( p/λ)2 + f ( p/λ).
λ[1 − F( p/λ)] − x j
0<
For the inverse hazard, we have h ( p/λ) < 0, and this implies f ( p/λ)2 /[1 − F( p/λ)] + f ( p/λ) > 0. Noting that
2λ/[λ[1 − F( p/λ)] − x j ] > 1/[1 − F( p/λ)] is implied by q2i > 0, we see that the above condition for M Ri < 0 is
satisfied.
Finally, note that M Ri = c has a unique solution at q2i > 0 when x j < x̄; when x j ≥ x̄, the best response is q2i = 0.
To see that the best response (when positive) has a slope strictly greater than −1, we can apply a standard monotone
comparative statics argument to the equivalent problem of choosing q ≥ x j to maximize [P2 (q) − c](q − x j ). The
cross-partial in q and x j of this objective is just M Ri , which is positive (and equal to c) at the solution. Thus, q is strictly
increasing in x j , and from q2i = q − x j we obtain the desired slope property.
Q.E.D.
Proof of Lemma 2. By Lemma 1, the period-2 equilibrium is symmetric and q2i∗ /q2∗ = 1/2. Then (5) becomes
P2∗ (x) ≡ P2 (q2∗ + x) = c −
1 ∗ ∗
q P (q + x).
2 2 2 2
(A1)
Therefore,
P2∗ (x) = c +
1 λ − x − λF(P2∗ (x)/λ)
,
2
f (P2∗ (x)/λ)
(A2)
upon substituting for P2 (q2 + x) as implied by (3) and for q2∗ as implied by (2). P2∗ (x) is the fixed point to (A2); existence
follows directly from the parameter assumptions on λ and c and the inverse-hazard property. Since the inverse hazard
h( p) is continuous and differentiable, the Implicit Function Theorem implies that P2∗ (x) is continuous and differentiable.
Differentiating both sides of (A2) with respect to x, and evaluating at p = P2∗ (x)/λ, we have
P2∗ (x) =
−1 f ( p) + f ( p)2 P2∗ (x) + λ−1 [λ − x − λF( p)] f ( p)P2∗ (x) .
2 f ( p)2
(A3)
Collecting terms and rearranging, we find
P2∗ (x)
λ − x − λF( p)
λ f ( p)2
−1
3λ f ( p)2
+ f ( p) =
.
λ − x − λF( p)
f ( p)
(A4)
The first bracketed term on the left is clearly positive. As in the proof of Lemma 1, h ( p) < 0 implies that the second
term is also positive. Therefore, we have P2∗ (x) < 0.
It is a straightforward comparative static exercise, employing (A2) to make explicit the dependence of P2∗ (x) on the
parameter λ, to verify that P2∗ (x) is differentiable and strictly increasing in λ.
Q.E.D.
Proof of Lemma 3. We know from condition (ii) in the Definition for equilibrium that if x > 0, then p1 ≤ δ P2e ( p1 ).
Now suppose p1 < δ P2e ( p1 ). We claim that x = Q S ( p1 ) ≥ x̄ must hold. If not, we have q2∗ (x) > 0 and P2∗ (x) > c,
by Lemma 1. Consequently, buyers with v above P2∗ (x) will purchase the quantity q2∗ (x) in period 2. However, with
p1 < δ P2e ( p1 ) = P2∗ (x), all of these buyers strictly prefer to purchase the good in period 1, in contradiction of optimal
storage choices. Thus, x ≥ x̄ and all units consumed in period 2 are purchased in period 1, with no further sales in period
2. From x, we calculate that all buyers with v ≥ vx ≡ λF −1 (1 − x/λ) purchase the good for storage in period 1, and
x ≥ x̄ implies c ≥ vx . Since vx ≥ p1 /δ is implied by optimal storage, we see that c ≥ p1 /δ and the period-1 price is
below marginal cost. Clearly, this cannot hold in equilibrium; it implies an overall negative profit for each firm, as there are
positive sales in period 1 (x > 0) and no further sales in period 2, so either firm would be strictly better off not producing.
The second part of Lemma 3 follows directly from the above argument and Lemma 1.
Q.E.D.
Proof of Lemma 5. (i) Given q j , the choice of q i is equivalent to the problem of choosing a q ≥ q j . Rewriting the
objective function in terms of q and q j , (q i , q j ) becomes
ˆ
(q,
q j ) = [P1 (q) − c](q − q j ) + δ2 (Q S (P1 (q))).
© RAND 2005.
mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
ANTON AND DAS VARMA / 539
ˆ
Since (q,
q j ) has strictly increasing differences in q and q j , as follows from strictly decreasing period -1 demand,
we can apply the standard monotone comparative static result for supermodular functions since the domains for q and q j
are each closed real intervals. Thus, the set of best responses for the q choice, (q j ), is strongly increasing: for q̂ j > q j ,
we have q̂ ≥ q for each q̂ ∈ (q̂ j ) and q ∈ (q j ).
(ii) Suppose, to the contrary, that (q j ) = N (q j ). Then S (q j ) ⊂ (q j ) for some such q j . By construction of
S , we have q i + q j ≥ q K for any q i ∈ S (q j ). Similarly, for N we have q̂ i + q̂ j ≤ q K for any q̂ i ∈ N (q̂ j ) . Since
is strongly increasing, we must also have q i + q j ≤ q̂ i + q̂ j . Therefore, q i + q j = q K = q̂ i + q̂ j . Further, q i > q̂ i
must hold, since q j < q̂ j . Thus, aggregate output is at the date-1 demand kink in both cases. Now, observe that S (q j )
can have no element greater than q i = q K − q j , or else would fail to be strongly increasing. By the feasible set for
S , we then have S (q j ) = {q K − q j }. But q i = q K − q j is necessarily feasible for the N problem at q j . Therefore,
the payoff from any choice in N (q j ) is at least as large as that from S (q j ), and we must then have N (q j ) ⊂ (q j ),
which contradicts the initial hypothesis. The later claim in part (ii) of the Lemma follows directly, since the inequalities
become strict in the preceding argument.
(iii) This is analogous to the proof of property (ii).
Existence of an equilibrium follows directly from Tarski’s intersection point theorem (see, e.g., Theorem 2.6 in Vives
(1999)), since is strongly increasing.
Q.E.D.
Proof of Lemma 6. We will show that any best response in for i to q j = q B necessarily involves an output strictly above
q K , implying strictly positive storage, when the (DS) condition in the lemma holds. Thus, by continuity of the payoff
function , this remains true for an interval where q j < q B . Therefore, we have q A < q B .
First, rewrite (q i , q j ) with price as the choice variable for firm i: choose p ≤ δp2C to maximize
S ( p, q j ) ≡ ( p − c)[Q 1 ( p) − q j ] + δ2 (Q S ( p)).
Clearly,
∂
S ( p, q j ) = ( p − c)Q 1 ( p) + [Q 1 ( p) − q j ] + δ2 (Q S ( p))Q S ( p)
∂p
= [1 − F( p) + Q S ( p) − q j ] − ( p − c) f ( p) + Q S ( p)[( p − c) + δ2 (Q S ( p))],
upon substituting for period-1 demand from (10). Next, from 2 (x) = [P2∗ (x) − c] 12 q2∗ (x), we have
1
1
1
2 (x) = [P2∗ (x) − c] q2∗ (x) + P2∗ (x) q2∗ (x) = −[P2∗ (x) − c] 1 + q2∗ (x) ,
2
2
2
where the last step follows from the definition P2∗ (x) = P2 (q2∗ (x) + x), which implies P2∗ (x) = P2 (q2∗ (x) + x)[q2∗ (x) + 1],
and from (5) for the period-2 equilibrium, which implies P2∗ (x) − c = −(1/2)q2∗ P2 (q2∗ (x) + x). Noting that P2∗ (x) = p/δ
by Lemma 3 for this price range, collecting terms and simplifying yields
∂
1
S ( p, q j ) = [1 − F( p) + Q S ( p) − q j ] − ( p − c) f ( p) + Q S ( p) ( p − c) − ( p − δc) 1 + q2∗ (x) .
∂p
2
Now, consider q j = q B and the choice p̂ ≡ δp2C ; note that this corresponds to a quantity choice by i of q i = q K − q B
(the kink in period-1 demand). By construction, ϕC (q B ) + q B = q K and we have f ( p̂)( p̂ − c) = 1 − F( p̂) − q j and
Q S ( p̂) = 0. Hence,
∂
1
= Q S ( p̂) ( p̂ − c) − ( p̂ − δc) 1 + q2∗ (0) .
S ( p, q j )
∂p
2
( p̂,q B )
Since Q S ( p) < 0, this expression is negative if and only if the second term is positive. Simplifying with α as defined
in the lemma yields the (DS) condition.
To complete the argument, suppose now that (DS) holds. This directly implies that all elements of (q B ) are strictly
greater than q K − q B . By construction, N (q B ) = {q K − q B } is the optimal choice for i subject to q i ≤ q K − q B . Every
optimal choice subject to q i ≥ q K − q B is strictly greater than q K − q B , since (q i , q B ) is strictly increasing in q i at
q i = q K − q B , as follows from
∂
< 0.
S ( p, q j )
∂p
( p̂,q B )
Further, since (q i , q j ) is continuous, the value of (q i , q j ) at q i ∈ (q B ) and q j = q B − ε, for small enough ε > 0,
must strictly exceed that value of (q i , q j ) at q i ∈ N (q B − ε) and q j = q B − ε. This establishes that q A < q B .
Q.E.D.
© RAND 2005.
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THE RAND JOURNAL OF ECONOMICS
Proof of Proposition 1. Consider the set of δ and λ such that (DS) is satisfied. From the definition P2∗ (x) = P2 (q2∗ + x),
we have P2∗ (x) = −[1 + q2∗ (x)]/ f (P2∗ (x)/λ). Then, P2∗ (x) from (A4) directly implies that 1/2 < α < 3/4. Hence, a
sufficient condition for (DS) is δ > (4c)/[ p2C + 3c]. This holds with strict inequality at δ = λ = 1, since the Cournot price
satisfies p2C > c. Since the Cournot price varies continuously with λ, (DS) continues to hold for a range of δ and λ values
strictly below one.
Now consider possible equilibria when δ = λ = 1. Note that p1C = δp2C in this case and the static Cournot outcome
for period 1 coincides with the kink in period-1 demand. Further, this is the only possible equilibrium with no storage.
We know, however, that q A < q B = q1C /2 holds and so either firm has a strict incentive to increase output and induce
/ (q1C /2). By Lemma 5, however, an equilibrium exists and, hence, it
storage, as Lemma 5 implies q1C /2 = ϕC (q1C /2) ∈
must have strictly positive storage.
Q.E.D.
Proof of Lemma 7. This applies for all δ ∈ [0, 1]. First, we dispense with the trivial case of extremely low discount factors,
δ ≤ c/ p2C , where the demand kink occurs at or below marginal cost. This implies q B (δ) = q K (δ) = 1− F(δp2C ) ≥ 1− F(c).
A simple dominance argument for the best response of i to q j then shows that reduces to the static Cournot best response,
ϕC , since it is never optimal to produce beyond the demand kink. Thus, q A (δ) = q K (δ) in this region.
Henceforth, we take δ ≥ c/ p2C . To make explicit the dependence of the problem on δ, we proceed as follows. Define
the value function VN (q j , δ) = maxq i ∈D N (q j ,δ) (q i , q j , δ), where the constraint set D N (q j , δ) ≡ [0, q K (δ) − q j ]; we
have q j ∈ [0, q K (δ)] and δ ∈ [c/ p2C , 1] for this problem. Similarly, VS (q j , δ) = maxq i ∈D S (q j ,δ) (q i , q j , δ) where the
constraint set D S (q j , δ) ≡ [max{q K (δ) − q j , 0}, 1 + λ]; we have q j ∈ [0, 1 + λ] and δ ∈ [c/ p2C , 1]. By the Maximum
Theorem (see, e.g., Sundaram, 1996), each value function is continuous over the associated set of (q j , δ).
Further, the standard envelope theorem applies whenever the optimal q i choice is interior to the constraint interval.
Thus, whenever interior choices are optimal we have
∂
VN (q j , δ) = 2 (0),
∂δ
∂
p
VS (q j , δ) = 2 (Q S ( p)) − {( p − c) f ( p) − [1 − F( p) + Q S ( p) − q j ]},
∂δ
δ
where p in the latter expression is the resulting price at an optimal interior quantity choice.
Case 1. Suppose we have 0 < q A (δ) < q B (δ) at δ. From above, δ > c/ p2C necessarily holds. By definition of q A and the
continuity of VS and VN , we see that
VS (q A (δ), δ) = VN (q A (δ), δ)
⇒
δ[2 (Q S ( p S )) − 2 (0)] = ( p N − c)q N − ( p S − c)q S ,
where ( p S , q S ) and ( p N , q N ) refer to price and an optimal quantity choice for each of VS and VN , respectively. Since
q A (δ) < q B (δ), we know that N (q j , δ) = {ϕC (q j )}, and this optimal choice is interior for any q j < q B (δ). Hence,
the N (q A (δ), δ) choice is interior and VN (q A (δ), δ) > (q K (δ) − q A (δ), q A (δ), δ). Any optimal choice in S (q A (δ), δ)
is therefore interior, since the corner choices for q i in D S (q j , δ) ≡ [q K (δ) − q A (δ), 1 + λ] cannot yield VS (q A (δ), δ) =
VN (q A (δ), δ), as required. Then
∂ = ( p N − c)q N − ( p S − c)q S + p S [q S − ( p S − c) f ( p S )],
VS (q j , δ) − VN (q j , δ) j
δ
q =q A (δ)
∂δ
upon substituting with the expression for VS (q A (δ), δ) = VN (q A (δ), δ) and with q S +q A = Q 1 ( p S ) = 1− F( p S )+ Q S ( p S ).
By interior choices, we know p N > δp2C > p S , Q S ( p S ) > 0, and q N < q S .
We show first that q S − ( p S − c) f ( p S ) > Q S ( p S ) > 0. To see this, note that [1 − F( p) − q A ]/ f ( p) − ( p − c) is
strictly decreasing in p, by the hazard assumption, and that it equals zero at the price p N = F −1 (1 − ϕC (q A ) − q A ). Since
p N > δp2C > p S , we then have [1 − F( p S ) − q A ]/ f ( p S ) − ( p S − c) > 0. Noting that q S = 1 − F( p S ) + Q S ( p S ) − q A
and that Q S ( p S ) > 0, the claim follows.
Next, we show ( p N − c)q N − ( p S − c)q S > −( p S − c)Q S ( p S ). Since q N = ϕC (q A ), we know
( p N − c)q N = max ( p − c)[1 − F( p) − q A ] > ( p S − c)[1 − F( p S ) − q A ] = ( p S − c)[q S − Q S ( p S )],
p
and the claim follows directly. Combining these two claims, we then have
∂
[VS (q A (δ), δ) − VN (q A (δ), δ)] > cQ S ( p S ) > 0.
∂δ
To see that q A (δ) is continuous and strictly decreasing at a δ where 0 < q A (δ) < q B (δ), simply note that we can
apply the Implicit Function Theorem to VS (q A (δ), δ) − VN (q A (δ), δ) since, by interior choices, we have
∂ VS (q j , δ) − VN (q j , δ) j
= p N − p S > 0,
j
q =q A (δ)
∂q
and, therefore, q A (δ) is differentiable with q A (δ) < −cQ S ( p S )/( p N − p S ) < 0.
© RAND 2005.
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ANTON AND DAS VARMA / 541
Case 2. Suppose q A (δ) = 0 at some δ. Again, this implies δ > c/ p2C . By definition of q A and continuity of (VS − VN ),
we must have VS (0, δ) − VN (0, δ) ≥ 0. If this is strict, then continuity also implies q A (δ̂) = 0 for any δ̂ sufficiently close
to δ, as implied by strict inequality of VS − VN and Lemma 5. Thus q A (δ) is continuous and weakly decreasing.
If equality holds, then we can argue as above that we must have interior choices for each of VS and VN at (0, δ).
Applying the envelope theorem as above, we conclude that (VS − VN ) is increasing in δ at (0, δ) and, hence, for δ̂
sufficiently close to δ that q A (δ̂) = 0 for δ̂ > δ and that q A (δ̂) > 0 for δ̂ < δ. Thus, q A (δ) is weakly decreasing in δ.
For continuity as δ̂ < δ approaches δ, consider limδ̂↑δ q A (δ̂). By the interior case, q A (δ̂) is positive and strictly
decreasing in δ̂, so the limit exists. Suppose the limit is positive and equals q̄ > 0. Since VS − VN is strictly increasing
in q j for the interior case, we know VS (q̄, δ) − VN (q̄, δ) ≡ ε > 0 must hold. By continuity of VS − VN , we can find
a γ > 0 for ε such that if (q j , δ̂) is within a radius of γ of (q̄, δ), then the corresponding VS − VN differences must be
within ε of each other. But then [VS (q A (δ̂), δ̂) − VN (q A (δ̂), δ̂)] must be positive for δ̂ near δ. But since q A (δ̂) is positive,
we know by construction that the difference is identically zero. Hence, q̄ = 0 must hold and we have continuity.
Case 3. Suppose q A (δ) = q B (δ) at δ. Consider continuity. If q A (δ) is not continuous at δ, then we can find a sequence
(δn ) converging to δ and an associated sequence (qn ) where qn ≡ q A (δn ) such that (qn ) converges to some q̄ < q A (δ).
Note that we can only have q̄ < q A (δ), since q A (δn ) ≤ q B (δn ) holds for all n. By definition of q A (δ), we must have
VS (q̄, δ) − VN (q̄, δ) < 0, since q̄ < q A (δ). Also, since qn ≡ q A (δn ), we must have VS (qn , δn ) − VN (qn , δn ) ≥ 0 for each
n and, therefore, limn→∞ [VS (qn , δn ) − VN (qn , δn )] ≥ 0. But this limit must coincide with VS (q̄, δ) − VN (q̄, δ), which is
negative, by continuity of VS − VN and the convergence of (δn ) to δ and (qn ) to q̄. Hence, we have a contradiction and,
therefore, q A (δ) is continuous at δ.
Consider monotonicity of q A (δ) at δ. If δ̂ > δ, then q A (δ̂) ≤ q B (δ̂) together with q B strictly decreasing imply
q A (δ̂) < q A (δ). On the other side, suppose there is some δ̂ < δ with q A (δ̂) ≤ q A (δ). We claim this implies that q A is
strictly decreasing on the interval [δ̂, δ) and, hence, that q A is not continuous at δ. To see why, note that q A (δ̂) < q B (δ̂)
must hold, since q B is strictly decreasing. Then for any δ ∈ (δ̂, δ) we must have q A (δ ) < q A (δ̂), since we can apply the
above monotonicity argument to δ > δ̂. Further, q A must be strictly decreasing at any such δ , since we have q A (δ ) <
q A (δ̂) ≤ q A (δ) = q B (δ) < q B (δ ). Thus, q A is strictly decreasing on [δ̂, δ) and we have limδ ↑δ q A (δ ) < q A (δ̂) ≤ q A (δ),
Q.E.D.
contradicting continuity of q A at δ. Hence, we must have q A (δ̂) > q A (δ) for δ̂ < δ and q A is strictly decreasing.
Proof of Proposition 2. Note that a no-storage equilibrium corresponds to the intersection of the 45◦ line with ∩ N .
Recall also that the only candidate for a no-storage equilibrium is the static Cournot equilibrium. We know from Proposition
1 that, with λ = 1 and δ sufficiently large, we have q A (δ) < q B (δ). Thus, an equilibrium with no storage cannot exist if
δ is sufficiently close to one. Also, from the proof of Lemma 7, we see that q A (δ) = q K (δ) > q1C /2 when δ < c/ p1C .
Define δ n ≡ [q A ]−1 (q1C /2). Observe that since q A (δ) is continuous and strictly decreasing in δ whenever it is positive, δ n
as defined above is unique. To prove the proposition, note that for any δ < δ n , q A (δ) > q1C /2, implying an intersection
of ∩ N with the 45◦ line. Therefore, for any δ < δ n , the no-storage equilibrium exists. Now consider any δ > δ n ,
which implies q A (δ) < q1C /2. This implies that (q1C /2) ∩ N (q1C /2) = ∅: the global best response to a rival’s output
must necessarily induce storage, when the latter is equal to the per-firm static Cournot (no storage) output. Therefore, the
no-storage equilibrium does not exist.
Q.E.D.
Proof of Proposition 3. Recall that
S (q j ) =
argmax
max{q K −q j ,0}≤q i ≤1+λ
P1 (q i + q j ) − c qi + δ2 (x),
(A5)
where P1 is the inverse of the aggregate-demand function, as given by the second line in (13), and x is as given by (12).
We know the storage best response, S (q j ), must have p1 > c, or else i would be better off reducing output. Restricting
attention to interior solutions of (A5), simplifying the first-order condition yields
S (q j ) = −
1+δ
2
qj +
1+δ
2
(1 + λ + 2c) +
c
(2δ − 1)(3 + δ).
2δ
By inspection, S (q j ) is single valued and strictly decreasing in q j . Furthermore, the partial derivative with respect
to δ is equal to
(1 + λ + 2c − q j )/2 + c
d
{(1 − 1/2δ)(3 + δ)}.
dδ
With q j < 1 + λ, this expression is positive. Hence S (q j ) is strictly increasing in δ.
Let ϕ S (q j , δ) denote the single-valued correspondence S . Using ϕ S , define δ s ≡ inf{δ | q A (δ) = ϕ S (q A (δ), δ)}.
We claim that a storage equilibrium exists if and only if δ ≥ δ s . Recall that storage equilibria correspond to points of
intersection of the 45◦ line with ∩ S . From above, we have that ϕ S (q A (δ), δ) is strictly increasing in its second argument.
Furthermore, it is strictly decreasing in its first argument. Since q A (δ) is decreasing in δ, we have that ϕ S (q A (δ), δ) is
strictly increasing in δ. To prove the claim, note then that for any δ < δ s , there cannot be a point of intersection between
© RAND 2005.
mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
542
/
THE RAND JOURNAL OF ECONOMICS
∩ S and the 45◦ line. On the other hand, for any δ ≥ δ s , there must exist such a point of intersection. Since ϕ S is
strictly decreasing with respect to q j , storage equilibrium must be unique.
Thus far, we have proved (i) and (iii). Since any equilibrium must be characterized by either storage or no storage,
by Lemma 5 (existence), we must have that δ s ≤ δ n . Then it remains to show that δ s < δ n . Now, q A (δ n ) ≡ q1C /2 <
ϕ S (q1C /2, δ n ) = ϕ S (q A (δ n ), δ n ), where the inequality follows from the fact that for any δ > c/ p1C , q1C = q1C /2 +
ϕC (q1C /2) < q K (δ) < q1C /2 + ϕ S (q1C /2, δ). But, as argued earlier, ϕ S (q A (δ), δ) is strictly increasing and continuous in
δ, whereas q A (δ) is decreasing and continuous in δ . Therefore, the two functions must cross at some value of δ (defined
δ s ), which is strictly below δ n .
Q.E.D.
Proof of Proposition 4. To begin, note that the (NS) problem reduces to a static monopoly choice with price constrained
to be above δ P2M (0). Since P1M satisfies this constraint (by assumption), it is the optimal choice and the payoff is
1M = (P1M − c)Q 1M . Now consider the (S) problem. Let π M (x) denote the objective function. First, note that for any
x > 0 we have
1M = P1M − c Q 1M > δ P2M (x) − c Q C1 δ P2M (x) ,
since p1 = δ P2M (x) is feasible but strictly suboptimal for the unconstrained static period-1 monopoly problem. Next,
noting that δ < 1, we clearly have [δ P2M (x) − c] < δ[P2M (x) − c]. Hence, we see that
π M (x) < 1M + δ P2M (x) − c x + δ2M (x).
Next, we claim that δ[P2M (x) − c]x + δ2M (x) ≤ δ2M (0). Consider the period-2 monopoly problem when we set
storage to zero. Note that q = x + Q 2M (x) is a feasible quantity choice for that problem. Also, from (3), the expression for
residual demand in period 2, we know P2M (x) = P2 (x + Q 2M (x)). Thus,
2M (0) = P2M (0) − c Q 2M (0)
≥ P2 x + Q 2M (x) − c [x + Q 2M (x)]
= 2M (x) + P2M (x) − c x,
as claimed. Combining the above bounds, static monopoly prices are optimal since
π M (x) < 1M + δ P2M (x) − c x + δ2M (x)
≤ 1M + δ2M (0).
Q.E.D.
Proof of Lemma 8. Consider monopoly. If storage is positive in equilibrium, then we necessarily have p1 ≤ δp2e . If
this is strict, then storage demand must be 1 − F( p1 /δ). In equilibrium, p2∗ = p2e must hold and, thus, the monopolist
would sell a quantity of zero and earn zero profit in period 2. But period-2 demand at prices below p1 /δ follows
Q C2 ( p2 ) = F( p1 /δ) − F( p2 ), and such prices yield a positive profit in period 2. Thus, the inequality cannot be strict and
we must have p1 = δ P2e ( p1 ) = δp2∗ when storage is positive.
Suppose p1 < δ P2M (0) and storage is zero. Then period-2 demand reduces to Q C2 ( p2 ) = 1 − F( p2 ), which coincides
with static demand, and the monopolist will set p2 = P2M (0). Since p2e = P2M (0) must hold in equilibrium, we then have
p1 < δp2e = δ P2M (0) and storage demand must be positive, which contradicts the hypothesis.
Now suppose p1 > δ P2M (0) and storage is positive. Since storage is positive, we know that p2∗ = p1 /δ must be
an optimal choice for the monopolist. Hence, p2∗ > P2M (0). Consider the period-2 payoff function for the monopolist
(suppress the dependence on p1 ):
π2M ( p2 ) = Q 2R ( p2 )[ p2 − c] =
[G(1) − F( p2 )]( p2 − c)
for p2 ≤ p1 /δ
[G(1) − G( p2 )]( p2 − c)
for p2 > p1 /δ.
For prices p2 ≤ p1 /δ, we see that π2M ( p2 ) = −[1 − G(1)]( p2 − c) + [1 − F( p2 )]( p2 − c). The first term is linear
and strictly decreasing in p2 . The second term is the static monopoly payoff function and we know it is strictly quasiconcave with a unique maximum at P2M (0). Since p1 /δ > P2M (0), the second term is strictly decreasing at p2 = p1 /δ.
Combining, we see that reducing p2 from p1 /δ must increase profit. Hence, it cannot be optimal for the monopolist to
charge p2∗ = p1 /δ and, therefore, storage must be zero when p1 > δ P2M (0).
The proof for the duopoly case is analogous and therefore omitted.
Q.E.D.
Proof of Proposition 5. From Lemma 8, the result is established if we can show that Q S ( p1 ) = [P M ]−1 ( p1 /δ) for
© RAND 2005.
mss # Anton & Das Varma; art. # 03; RAND Journal of Economics vol. 36(3)
ANTON AND DAS VARMA / 543
p1 ≤ δ P2M (0). In equilibrium, it must be optimal for the monopolist to set a period-2 price of p1 /δ for period-2 demand
of Q 2R ( p2 ) under the rationing rule R. Recall π2M ( p2 ) from the proof of Lemma 8. We know that π2M ( p2 ) is differentiable
for p2 ≤ p1 /δ, and direct calculation yields
d M
π ( p2 ) = −[1 − G(1)] + [1 − F( p2 )] − f ( p2 )]( p2 − c) ⇒
d p2 2
d M
π2 ( p2 )
= −[1 − G(1)] + [1 − F( p1 /δ)] − f ( p1 /δ)]( p1 /δ − c) ≥ 0,
d p2
p2 = p1 /δ
which is necessary for an optimum. With p1 ≤ δ P2M (0), we know that the static monopoly payoff is strictly increasing
in this price range. Now consider p2 ≥ p1 /δ and note that we now have
π2M ( p2 ) = [G(1) − G( p2 )]( p2 − c)
= −[1 − G(1)]( p2 − c) + [1 − F( p2 )]( p2 − c) + [F( p2 ) − G( p2 )]( p2 − c).
Optimality requires that π2M ( p2 ) be nonincreasing at p2 = p1 /δ. The third term is always nonnegative for any R and
it is zero at p2 = p1 /δ. Thus, π2M ( p2 ) would be strictly increasing if the first two terms, which are differentiable, were
strictly increasing. But these terms are the same as in the previous case and we can immediately conclude that optimality
of p2 = p1 /δ requires −[1 − G(1)] + [1 − F( p1 /δ)] − f ( p1 /δ)]( p1 /δ − c) ≤ 0. Thus, in equilibrium we must have
G(1) = 1 − F( p1 /δ) − f ( p1 /δ)( p1 /δ − c),
for any rationing rule. But this is exactly the definition of [P M ]−1 ( p1 /δ) and, thus, we are done.
The duopoly case is again analogous and the proof is omitted.
Q.E.D.
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